Hypercyclic behaviour of multiples of composition operators on weighted Banach spaces of holomorphic functions (Q464208)

For the weight \(v(z)=(1-|z|^2)^{\alpha}\), \(\alpha>0,\) on the unit disc \(\mathbb{D}\) of the complex plane, the authors consider the space \(H^{\infty}_{\alpha,0}\) of all the analytic functions \(f \in H(\mathbb{D})\) such that \(\lim_{|z| \rightarrow 1} v(z) |f(z)| =0\) and they complement the work \textit{A. Miralles} and \textit{E. Wolf} [Math. Nachr. 286, No. 1, 34--41 (2013; Zbl 1294.47008)] by studying when operators of the form \(T= \lambda C_{\varphi}\), \(\lambda \in \mathbb{C}\), \(\lambda \neq 0,\) for a self map \(\varphi\) on \(\mathbb{D}\), are hypercyclic. Among other results, they prove: (1) If \(\varphi\) has an interior fixed point, then \(T\) is not hypercyclic. (2) Characterizations for hypercyclic \(T\) are given when \(\varphi\) is a parabolic or hyperbolic automorphism. (3) A characterization of hypercyclic \(T\) when \(\varphi\) is a hyperbolic non-automorphism. (4) Some results about frequent hypercyclicity of \(T\).